A note on Artico-Moresco fuzzy proximity spaces (Q2735529)

Let \({\mathbf A}\) be a fibre-small topological construct and \(X\) a set. A co-tower of \({\mathbf A}\)-structures on \(X\) is a function \(\Gamma\) from \([0,1]\) to the complete lattice of \({\mathbf A}\)-structures on \(X\) such that NEWLINE\[NEWLINE\{(X,\Gamma (\alpha)) @>\text{id}_X>> (X,\Gamma (\beta))\}_{\beta >\alpha}NEWLINE\]NEWLINE is an initial source for each \(\alpha\in [0,1]\). A morphism between co-tower spaces \((X,\Gamma)\to(Y,\Xi)\) is a function \(f:X\to Y\) such that \(f:(X,\Gamma (\alpha)) \to(Y,\Xi (\alpha))\) is a morphism in \({\mathbf A}\) for each \(\alpha\in I\). The construct of co-tower spaces and morphisms is denoted \({\mathbf A}^c(I)\), called the co-tower extension of \({\mathbf A}\).NEWLINENEWLINENEWLINEIt is known that the topological construct of fuzzy neighborhood spaces [\textit{B. Lowen}, Fuzzy Sets Syst. 7, 165-189 (1982; Zbl 0487.54008)] and that of Lowen fuzzy uniform spaces [\textit{R. Lowen}, J. Math. Anal. Appl. 82, 370-385 (1981; Zbl 0494.54005)] are isomorphic to the co-tower extensions of the construct of topological spaces and uniform spaces respectively. In this paper, the authors add another example to this list, precisely, they show that the topological construct of Artico-Moresco fuzzy proximity spaces [\textit{G. Artico} and \textit{R. Moresco}, Fuzzy Sets Syst. 21, 85-98 (1987; Zbl 0612.54006); 31, No. 1, 111-121 (1989; Zbl 0696.54007)] is isomorphic to the co-tower extension of the topological construct of proximity spaces.